Answer to a question of Hung and Tiep on conductors of cyclotomic integers

Abstract

In Question 5.2, Hung and Tiep (Eur J Math 9, 2023) asked the following: If \(\alpha \) is a sum of k complex roots of unity and \(\mathbb {Q}_{c(\alpha )}\) is the smallest cyclotomic field containing \(\alpha \), is it true that \(|\mathbb {Q}_{c(\alpha )}:\mathbb {Q}(\alpha )| \le k\)? We answer this question in the negative, and in §4, we shall bound the growth of \(|\mathbb {Q}_{c(\alpha )}:\mathbb {Q}(\alpha )|\) as a function of k using known results on minimal vanishing sums.